One Parameter Groups Associated with Quantum Girsanov Transformation (Mathematical aspects of quantum fields and related topics)

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(1)21 21. One Parameter Groups Associated with Quantum Girsanov Transformation. Mi Ra Lee. Department of Mathematics Chungbuk National University. 1. Preliminary. Let T be a topological space with Borel measure dt and H=L^{2}(T, dt) be a Hilbert space equipped with a norm | . |_{0} . For a positive self‐adjoint operator A on H with Hilbert Schmidt inverse and each p\geq 0 , define. E_{p}=\{\xi\in H:|\xi|_{p}=|A^{p}\xi|_{0}<\infty\} and E_{-p} by closure of. H. with respect to a norm |\xi|_{-p} :=|A^{-p}\xi|_{0}, \xi\in H . Then we have. a chain of Hilbert spaces as. . . . \subset E_{p}\subset H\subset E_{-p}\subset By defining E=. proj 1. \dot{ \imath} mE_{p}par ow\infty, E^{*}= ind\lim_{arrow p\infty}E_{-p}. the Gelfand triple is obtained: E\subset H\subset E^{*}. The canonical bilinear form on Fock spaces of E_{p} given by. E^{*}\cross E. is denoted by \langle\cdot, \cdot\rangle . For each p\geq 0 , the Boson. \Gamma(E_{p})=\{\phi=(f_{n})_{n=0}^{\infty}:\Vert\phi\Vert_{p}=\sum_{n=0} ^{\infty}n!|f_{n}|_{p}<\infty, f_{n}\in E_{p}^{\otimes^{\wedge}n}\} derive a chain of Fock spaces. \subset\Gamma(E_{p})\subset\Gamma(H)\subset\Gamma(E_{-p})\subset Define. (E)= proj 1. \dot{ \imath} m\Gamma(E_{p})par ow\infty, (E)^{*}= ind\lim_{arrow p\infty}\Gamma(E_{-p}). then we again have the Gelfand triple. (E)\subset\Gamma(H)\subset(E)^{*} In particular, it is known as Hida−Kubo−Takenaka space with the Wiener‐Itô‐Segal iso‐. morphism L^{2}(E^{*}, \mu)\cong\Gamma(H) where E^{*} .. \mu. is a standard Gaussian probability measure on. Note that the set of exponential vectors. \phi_{\xi}=(1, \xi, \cdots, \frac{1}{n!}\xi^{\otimes n}, \cdots) , \xi\in E.

(2) 22 spans a dense subspace of (E) . The topology of (E) is given by the norm. \Vert\phi\Vert_{p}^{2}=\sum_{n=0}^{\infty}n!|f_{n}|^{2}, \phi=(f_{n})_{n=0} ^{\infty}, p\in \mathb {R}. Moreover, for \Phi\in(E)^{*} , there exists p\geq 0 such that \Phi\in\Gamma(E_{-p}) , that is. \Vert\Phi\Vert_{-p}^{2}=\sum_{n=0}^{\infty}n!|F_{n}|_{-p}^{2}<\infty, \Phi=(F_ {n})_{n=0}^{\infty}. The canonical bilinear form on (E)^{*}\cross(E) is given by. \langle\langle\Phi, \phi\rangle\}=\sum_{n=0}^{\infty}n!\langle F_{n}, f_{n}\}, \Phi=(F_{n})_{n=0}^{\infty}, \phi=(f_{n})_{n=0}^{\infty}. 2. White Noise Operators. An operator ---\in \mathcal{L}((E), (E)^{*}) is called a white noise operator where \mathcal{L}((E), (E)^{*}) is the. space of continuous linear operator from (E) to (E)^{*} operator a(x)\in \mathcal{L}((E), (E)) is given by. For x\in(E)^{*} , the annihilation. a(x):\phi=(f_{n})_{n=0}^{\infty}\mapsto((n+1)x\otimes_{1}f_{n+1})_{n=0} ^{\infty} where x\otimes_{1}f_{n} is a contraction of tensor product. The adjoint of annihilation operator a^{*}(x)\in \mathcal{L}((E)^{*}, (E)^{*}) is called a creation operator, and its action is given by. a^{*}(x):\phi=(f_{n})_{n=0}^{\infty}\mapsto(x\otimes f_{n-1})_{n=0}^{\infty} \wedge. For. \kappa\in(E^{\otimes(l+m)})^{*} and \phi=(f_{n})_{n=0}^{\infty}\in(E) , we define a sequence (g_{n})_{n=0}^{\infty} by. g_{n}=0,0 \leq n<l, g_{l+n}=\frac{(n+m)!}{n!}\kappa\otimes_{m}f_{n+m}, n\geq 0. The operator \Xi_{l,m} defined by \Xi_{l,m}\phi=(g_{n})_{n=0}^{\infty} is called the integral kernel operator with kernel distribution \kappa . The following theorem shows that each white noise operator has unique infinite series expansion called a Fock expansion.. Theorem 2.1 (Obata [12]) For each ---\in \mathcal{L}((E), (E)^{*}) , there exists a unique family of. \kappa_{l,m}\in(E^{\otimes(l+m)})^{*}. such that. ----(K_{l,m}) whenever the sum converges in \mathcal{L}((E), (E)^{*}) . For S\in \mathcal{L}(E, E^{*}) associated distribution \tau_{S}\in(E\otimes E)^{*} is given by. \langle\tau_{S}, \xi\otimes\eta\rangle=\langle S\eta, \xi\}, \xi, \eta\in E..

(3) 23 If \langle S\eta, \xi\rangle=\langle S\xi, \eta\} , i.e. S=S^{*} , then S is called symmetric. orthonormal basis of H , then \tau_{S} has infinite series expansion. If \{e_{n}\}_{n=0}^{\infty} is an. \tau_{S}=\sum_{n=0}^{\infty}Se_{n}\otimes e_{n}. The quadratic annihilation operator associated with S is defined by \triangle_{G}(S)=\Xi_{0,2}(\tau_{S}) . Applying to the exponential vector \phi_{\xi} of \xi\in E , its action is understood by. \triangle_{G}(S)\phi_{\xi}=\{S\xi, \xi\rangle\phi_{\xi}. Moreover, \triangle_{G}(S) has infinite series expansion. \triangle_{G}(S)=\sum_{n=0}^{\infty}a(Se_{n})a(e_{n}). .. The dual of \triangle_{G}(S) with respect to the canonical bilinear form is called the quadratic creation operator. \triangle_{G}^{*}(S)=\Xi_{2,0}(\tau_{S})=\sum_{n=0}^{\infty}a^{*}(Se_{n})a^{*} (e_{n}). .. The conservation operator \Lambda(S)\in \mathcal{L}((E), (E)^{*}) is defined by \Lambda(S)=\Xi_{1,1}(\tau_{S}) , and its infinite series expansion is obtained as. \Lambda(S)=\sum_{n=0}^{\infty}a^{*}(Se_{n})a(e_{n}). .. The second quantization of S is defined by. \Gamma(S)\phi=(S^{\otimes n}f_{n})_{n=0}^{\infty}, \phi=(f_{n}) and is related to the conservation operator as. \frac{d\Gamma(e^{tS})}{dt}|_{t=0}=\Lambda(S). .. \Lambda(S) is also called the differential second quantization operator.. 3. Quantum White Noise Differential Equations. For \Xi\in \mathcal{L}((E), (E)^{*}) and \zeta\in E , a commutator. [a(\zeta), ---]=a(\zeta)_{--}^{--}---a(\zeta) , [a^{*}\zeta,---]=a^{*}(\zeta)_{ --}^{--}---a^{*}(\zeta) are well‐defined by composition since a(\zeta)\in \mathcal{L}((E), (E))\cap \mathcal{L}((E)^{*}, (E)^{*}) and a^{*}(\zeta)\in. \mathcal{L}((E)^{*}, (E)^{*})\cap \mathcal{L}((E), (E)) .. Define. D_{\zeta-}^{+-}-=[a(\zeta), ---], D_{\zeta^{-}}^{--}-=-[a^{*}\zeta, ---], which are called the creation and annihilation derivatives respectively. Both together are called quantum white noise derivatives..

(4) 24 (\zeta, \Xi)\mapsto D_{\zeta}^{\pm}\Xi. Theorem 3.1 (Ji‐Obata [6]) \mathcal{L}((E), (E)^{*}) to \mathcal{L}((E), (E)^{*}) .. is a continuous bilinear map from. E\cross. Example 3.2 (Ji‐Obata [8]) For S\in \mathcal{L}(E, E^{*}) and \zeta\in E we have. D_{\zeta}^{+}\triangle_{G}(S)=0, D_{\zeta}^{-}\triangle_{G}(S)=a(S\zeta)+ a(S^{*}\zeta) D_{\zeta}^{+}\triangle_{G}^{*}(S)=a^{*}(S\zeta)+a^{*}(S^{*}\zeta) , D_{\zeta}^{ -}\triangle_{G}^{*}(S)=0, D_{\zeta}^{+}\Lambda(S)=a(S^{*}\zeta) , D_{\zeta}^{-}\Lambda(S)=a^{*}(S\zeta). ,. .. For \Xi\in \mathcal{L}((E), (E)^{*}) , a function - \wedge on. E\cross E. defined by. - -(\xi, \eta)=\wedge\langle\langle\Xi\phi_{\xi}, \phi_{\eta}\rangle\rangle is called the operator symbol of. \Xi .. Note that the mapping \Xi\mapsto-\wedge-- is injective.. Proposition 3.3 (Obata [11]) Let \Theta be a function on E\cross E with values on \mathb {C} . Then there exists a continuous operator ---\in \mathcal{L}((E), (E)^{*}) such that \Theta=--\wedge- if and only if. (1). \Theta. is Gâteaux entire function;. (2) for any p\geq 0 and. \epsilon>0 ,. there exist C\geq 0 and q\geq 0 such that. |\Theta(\xi, \eta)|\leq C\exp\{\epsilon(|\xi|_{p+q}^{2}+|\eta|_{-p}^{2})\}, \xi, \eta\in E. Let \Xi_{1}, \Xi_{2}\in \mathcal{L}((E), (E)^{*}) . Then there exists a unique \Xi\in \mathcal{L}((E), (E)^{*}) satisfying. --(\xi, \eta)=e-(\xi, \eta)_{-2}^{-}(\xi, \eta) where -\wedge- is an operator symbol of H Then : is called the Wick product of --1- and --2and denoted by \Xi=\Xi_{1}0\Xi_{2} . For examples,. a(x)\Diamond a(y)=a(x)a(y) , a^{*}(x)\circ a^{*}(y)=a^{*}(x)a^{*}(y) a(x)oa^{*}(y)=a^{*}(y)a(x) , a^{*}(x)oa(y)=a^{*}(x)a(y). ,. .. The right hand sides of above examples are called the wick ordered form of given oper‐ ators. More generally, for ---\in \mathcal{L}((E), (E)^{*}) one has. a^{*}(x_{1})\cdots a^{*}(x_{2})\Xi a(y_{1})\cdots a(y_{m})=(a^{*}(x_{1})\cdots a^{*}(x_{2})a(y_{1})\cdots a(y_{m}))0\Xi. Equipped with Wick product, \mathcal{L}((E), (E)^{*}) becomes a commutative * ‐algebra. For Y\in \mathcal{L}((E), (E)^{*}) the wick exponential of Y is defined by wexp. Y= \sum_{n=0}^{\infty}\frac{1}{n!}Y^{on}. whenever the series converges in \mathcal{L}((E), (E)^{*}) . A continuous map \mathcal{D} : \mathcal{L}((E), (E)^{*})arrow \mathcal{L}((E), (E)^{*}) satisfying. \mathcal{D}(-0-)=(\mathcal{D}_{-1}^{---}-)\vartheta_{-2}^{-+_{-2}}- a(\mathcal{D}_{-2}^{-}-) is called a wick derivation..

(5) 25 Theorem 3.4 (Ji‐Obata [8]) The creation and annihilation derivatives are wick deriva‐ tions.. According to wick derivation, Ji‐Obata [8] introduced first order homogeneous linear differential equation of wick type.. Theorem 3.5 (Ji‐Obata [8]) Let G\in \mathcal{L}((E), (E)^{*}) . If there is an operator U\in \mathcal{L}((E), (E)^{*}) such that \mathcal{D}U=G and wexp U\in \mathcal{L}((E), (E)^{*}) ) then the solution of linear differential equation: \mathcal{D}\Xi=GQ\Xi. is given by \Xi=Fo. (wexp. where the operator F\in \mathcal{L}((E), (E)^{*}) satisfies. U). \mathcal{D}F=0.. For non‐homogeneous type, we refer [9]. Proposition 3.6 (Ji‐Obata [8]) Let \Xi\in \mathcal{L}((E), (E)^{*}) .. D_{\zeta-}^{+-}-=0 for all. (1). \zeta\in E if and only if − is of the form: --. \Xi=\sum_{m=0}^{\infty}\Xi_{0,m}(\kap a_{0,m}) (2) D_{\zeta}^{-}\Xi=0 for all \zeta\in E if and only if. \Xi. .. is of the form:. ----l,0-.. (3) If. \Xi. satisfies. D_{\zeta}^{+}\Xi=D_{\zeta}^{-}\Xi=0 for all. \zeta\in E,then\Xi is a scalar operator.. Example 3.7 Let \eta, \zeta\in E and S\in \mathcal{L}(E, E) . By solving wick type differential equa‐ tions, we can get the wick ordered form of : as following:. e^{a(\zeta)}e^{a^{*}(\eta)}=e^{\langle\zeta,\eta\rangle}e^{a^{*}(\eta)} e^{a(\zeta)} ,. (3.1). \Gamma(S)e^{a^{*}(\eta)}=e^{a^{*}(S\eta)}\Gamma(S) , e^{a(\zeta)}\Gamma(S)=\Gamma(S)e^{a(S^{*}\zeta)} .. (3.2). Indeed, for (3.1), take creation and annihilation derivatives for \xi\in E to. (3.3) \Xi ,. then we have. D_{\xi}^{+}\Xi=D_{\xi}^{+}e^{a(\eta)}e^{a^{*}(\zeta)}=e^{a(\eta)}(D_{\xi}^{+}e^ {a^{*}(\zeta)}) =e^{a(\eta)}(D_{\xi}^{+}a^{*}(\zeta) e^{a^{*}(\zeta)}=\langle\xi, \zeta\rangle e^{a(\eta)}e^{a^{*}(\zeta)}=\langle\xi, \zeta\rangle I<\rangle- -, D_{\xi}^{-}\Xi=D_{\xi}^{-}e^{a(\eta)}e^{a^{*}(\zeta)}=(D_{\xi}^{-}a(\eta))e^{a( \eta)}e^{a^{*}(\zeta)} =\langle\eta, \xi\}e^{a(\eta)}e^{a^{*}(\zeta)}=\{\eta, \xi\rangle I_{\vartheta_ {-}^{-} ^{-}. Let has. Y. satisfy. D_{\xi}^{+}Y=\langle\xi,. \zeta }. I. and D_{\xi}^{-}Y=\langle\eta, \xi } I . Then from creation derivative one. Y=a^{*}(\zeta)+Y_{1}, D_{\xi}^{+}Y_{1}=0.

(6) 26 and from annihilation derivative we have. D_{\xi}^{-}Y=D_{\xi}^{-}Y_{1}=\langle\eta, \xi\} which implies that Y_{1}=a(\eta)+Y_{2} where scalar operator C . Then \Xi=C . wexp. D_{\xi}^{-}Y_{2}=0 . So Y=a^{*}(\zeta)+a(\eta)+C for some Y=C\cdot e^{a^{*}(\zeta)}e^{a(\eta)}. and C is obtained by. C=\langle\langle\Xi\phi_{0}, \phi_{0}\rangle\rangle=\langle\langle e^{a^{*} (\zeta)}\phi_{0}, e^{a^{*}(\eta)}\phi_{0}\rangle\rangle=e^{\langle\zeta, \eta\rangle}. Similarly we can get (3.2) and (3.3). Furthermore, the wick ordered form of white noise operators including up to quadratic annihilation and creation operators are well known.. For more details see Ji‐Obata [9, 10].. 4. One Parameter Group. Motivated from [10], we construct one parameter group involving annihilation, creation and conservation operators. For a locally convex space X , let GL(X) be a group of linear homomorphisms in X . A one‐parameter family \{T_{\theta}\}_{\theta\in \mathbb{R} is called a group if (1) T_{0}=I ;. (2) T_{\theta_{1}+\theta_{2} =T_{\theta_{1} T_{\theta_{2} , for \theta_{1}, \theta_{2}\in \mathbb{R}. Let \eta,. \zeta be differentiable functions from. function from \mathbb{R} to. we put. \mathbb{R} with values on E and A be a differentiable. \mathcal{L}(E, E) . Let C be a differentiable function on \mathbb{R} . For each \theta\in \mathbb{R}. T_{\theta}=C(\theta)e^{a^{*}(\eta(\theta))}\Gamma(A(\theta))e^{a(\zeta(\theta)) }.. To satisfy the group conditions, we consider the compositions of T_{\theta_{1} and T_{\theta_{2}. T_{\theta_{1} T_{\theta_{2} =C(\theta_{1})C(\theta_{2})e^{a^{*}(\eta(\theta_{1} ) }\Gamma(A(\theta_{1}) e^{a(\zeta(\theta_{1}) }e^{a^{*}(\eta(\theta_{2}) } \Gamma(A(\theta_{2}) e^{a(\zeta(\theta_{2}) } =C(\theta_{1})C(\theta_{2})e^{\langle\zeta(\theta_{1}),\eta(\theta_{2})\rangle} e^{a^{*}(\eta(\theta_{1})+A(\theta_{1})\eta(\theta_{2}) }\Gamma(A(\theta_{1}) A(\theta_{2}) e^{a(\zeta(\theta_{2})+A(\theta_{2})^{*}\zeta(\theta_{1}) } by applying (3.1), (3.2) and (3.3). Then the group property T_{\theta_{1}+\theta_{2} =T_{\theta_{1} T_{\theta_{2} induces following equations:. \eta(\theta_{1}+\theta_{2})=\eta(\theta_{1})+A(\theta_{1})\eta(\theta_{2}) , A(\theta_{1}+\theta_{2})=A(\theta_{1})A(\theta_{2}) , \zeta(\theta_{1}+\theta_{2})=\zeta(\theta_{2})+A(\theta_{2})^{*} \zeta(\theta_{1}) ,. (4.1) (4.2) (4.3). C(\theta_{1}+\theta_{2})=C(\theta_{1})C(\theta_{2})e^{\langle\zeta(\theta_{1}), \eta(\theta_{2})\}. (4.4). and initial values are obtained as. A(0)=I, \eta(0)=\zeta(0)=0..

(7) 27 Proposition 4.1 The one parameter family \{T_{\theta}\}_{\theta\in \mathbb{R} is a group if E ‐valued functions \eta, \zeta, \mathcal{L}(E, E) ‐valued function A and real valued function C satisfy following differential equations:. \eta'(\theta)=A(\theta)\eta'(0) , A'(\theta)=A(\theta)A'(0) , \zeta'(\theta)=A(\theta)^{*}\zeta'(0) , C'(\theta)=C'(0)C(\theta) . PROOF.. (4.5) (4.6) (4.7) (4.8). Take \theta_{1}=\theta and \theta_{2}=h . Then from (4.2) we see A(\theta+h)-A(\theta)=A(\theta)[A(h)-I]. which shows (4.6). In the same way, by taking \theta_{1}=\theta and \eta(\theta+h)-\eta(\theta)=A(\theta)\eta(h). \theta=h ,. we get. .. Thus (4.5) is obtained. Similarly we have (4.7). For (4.8), consider a function f(x)=. C(x)e^{\langle\zeta(\theta),\eta(x)\rangle} .. Then. f. is differentiable and. f'(x)=C'(x)e^{\langle\zeta(\theta),\eta(x)\rangle}+C(x)\langle\zeta(\theta), \eta(x)\rangle e^{\langle\zeta(\theta),\eta(x)\}} f'(0)=C'(0) .. shows that. So. C'( \theta)=C(\theta)\lim_{har ow 0}\frac{C(h)e^{\langle\zeta(\theta),\eta(h) \rangle} {h}=C(\theta)C'(0) is obtained.. I. \eta=\eta'(0), \zeta=\zeta'(0)\in E, A=A'(0)\in \mathcal{L}(E, E) be given. The solutions of (4.5) -(4.8) are obtained as followings:. Theorem 4.2 Let. and. c=C'(0)\in \mathbb{R}. A(\theta)=e^{\theta A}, C(\theta)=e^{c\theta}. \eta(\theta)=\int_{0}^{\theta}e^{tA}\eta dt, \zeta(\theta)=\int_{0}^{\theta}e^ {tA^{*} \zeta dt.. The proof is straightforward. Theorem 4.3 \{T_{\theta}\}_{\theta\in \mathbb{R} is. a. one‐parameter group with the infinitesimal generator. \frac{dT_{\theta} {d\theta}|_{\theta=0}=cI+a^{*}(\eta)+\Lambda(A)+a(\zeta) where. \eta=\eta'(0), \zeta=\zeta'(0), A=A'(0), c=C'(0) .. PROOF.. We can see by applying characterization theorem of operator symbol of. T_{\theta} .. I. Corollary 4.4 Let A(\theta)=I for \theta\in \mathbb{R} . Then one parameter family of. T_{\theta}=C(\theta)e^{a^{*}(\eta(\theta))}e^{N}e^{a(\zeta(\theta))} is a group with the infinitesimal generator I+a^{*}(\eta)+N+a(\zeta) where \eta=\eta'(0), \zeta=\zeta'(0). and N=\Lambda(I) is a number operator. For more details, see [2]. As a general case, one parameter group with the infinitesimal generator which is. a linear combination of a^{*}(\eta), a(\zeta), \Lambda(B), \triangle_{G}(A) and \triangle_{G}^{*}(C) is studied in [4], where A, B, C,. \eta,. \zeta satisfy certain conditions..

(8) 28 References [1] Berezin, E.A. (1966), “The Method of Second Quantization. Academic Press.. [2] Chung, D.M. and Ji, U.C. (2000), Multi‐parameter transformation groups on white noise functionals, J. Math. Anal. Appl. 252, no.2, 729‐749.. [3] T. Hida (1975), “Analysis of Brownian Functionals. Carleton Math. Lect. Notes”. no. 13, Carleton University, Ottawa.. [4] Ji, U.C. and Lee, M.R. (2018), One‐parameter transformation groups on nuclaer rigging of Fock space, Preprint.. [5] Ji, U.C. and Obata, N. (2005), Admissible white noise operators and their quantum white noise derivatives, Infinite dimensional harmonic analysis , 213‐232, World Sci. Publ., Hackensack, NJ.. [6] Ji, U.C. and Obata, N. (2007), Generalized white noise operator fields and quantum white noise derivatives, Seminaires et Congrés, 16, 17‐33.. [7] Ji, U.C. and Obata, N. (2009), Annihilation‐derivative, creation‐derivative and rep‐ resentation of quantum martingales, Commun. Math. Phys., 286, 751‐775.. [8] Ji, U.C. and Obata, N. (2010), Implementation problem for the canonical commu‐ tation relation in terms of quantum white noise derivatives, J. Math. Phys., 51, 123507.. [9] Ji, U.C. and Obata, N. (2013), Calculating normal‐ordered forms in Fock space by quantum white noise derivatives Interdiscip. Inform. Sci., 19, 201‐211.. [10] Ji, U.C. and Obata, N. (2016), An implemetation problem for boson fields and quantum Girsanov transform, J. Math. Phys., 57, 083502.. [11] Obata, N. (1993), An analytic characterization of symbols of operators on white noise functionals, J. Math. Soc. Japan, 45, 421‐445.. [12] Obata, N. (1994), “White Noise Calculus and Fock Space Vol. 1577, Springer‐Verlag. Department of Mathematics Chungbuk National University Cheongju, 28644 Korea E ‐Mail: mrlee1012@chungbuk.ac.kr. Lect. Notes in Math.,.

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